Opening credits

My interpretation is that this is a kind of fever dream mutually shared by Karras and the Gemini Killer/Vennamun. There's no reason that Vennamun should remember "a fall down a long flight of stairs" - that's clearly Karras' memory of his own death. The vengeful demon does not place the Gemini into Karras until after the fall, thus too late for the Gemini to remember anything about it.

Otoh, there's a tapping sound like a walking cane meeting the pavement. Perhaps this suggests that the Gemini is out for a walk in the body of one of his infirm "old friends" while "his" (Karras's) body is still in a straitjacket in the hospital cell. Since both Karras's soul and the Gemini's soul are sharing the same body, there is every possibility that two separate thought-and-memory systems are in play, and the opening sequence may be expressing this idea by depicting them sharing a dream and reliving old memories.

The running priest: note the location of the boy with the rose and the running priest: directly in front of the MacNeil house from the first film, with the camera inevitably approaching the top of the Hitchcock Steps where Karras died. The opening places us directly and immediately at the core locations where Regan MacNeil was possessed, Burke Dennings was murdered, where Kinderman climbed the Steps finding evidence, where Merrin stood in front of the house, etc. We are brought back to the beginning, and the ending, of the first film. And now we see a cassocked priest running twice across the street in front of the MacNeil house.

Perhaps the priest is part of Karras's memory/fever dream... dreams are distortive, and Karras may be dreaming that he is back at the house/Steps again, running in a dream-panic.
Or maybe it's a shared Karras-Gemini memory of the very night of the MacNeil exorcism. Remember that Karras (the reanimated Karras-cum-Gemini personality) "was found wandering the C&O Canal" on the night of the exorcism. This indicates that after the Gemini caused the reanimated Karras to break out of his cheap coffin ("vow of poverty...disgusting!"), he wandered the Georgetown streets until he was finally picked up by authorities. This gives him plenty of time to run around, crazed, in the vicinity of his most recent living memory. Why "crazed"? Because - we can imagine - the resuscitation, plus the shock of both Karras and Vennamun being slammed back into "the body", plus the trauma of Vennamun's recent execution and Karras's recent fall-and-death, could cause "them", both in Karras's body, to run in a blind panic, before finally adjusting to their new embodiment, and finally calming down sufficiently to listlessly wander the C&O Canal.

The black boy "Thomas Kintry" holding the rose is more difficult to explain, since he is only known to Kinderman, and the opening is not Kinderman's dream or fantasy. However, it is plausible that Kintry was somewhat known to Vennamun, who, directing the hapless Karras's body, drugged, crucified and decapitated Kintry on the boathouse dock. So this may be Vennamun's, not Karras's, portion of the theoretical shared dream. Why he appears twice is a mystery, but of course, if we regard this as a dream, it is true that dreams, as mentioned above, are distortive. Karras/Vennamun simply see the boy twice under the strange weavings of the dream.

The rose is not explicable from any data Blatty gives us. There is nothing in Karras's or the Gemini's past that suggests roses as significant. Nor is there anything in Kintry's past to associate him with a rose - he's a newspaper deliverer, not a flower salesperson. Elsewhere, though, it is said that for Blatty the rose represents purity, as it did indeed in the first novel, where Regan MacNeil was in the habit of leaving a fresh rose at her mother's place at the breakfast table. But we as filmgoers were never told of the book's Regan-rose association, so its presence in Exorcist III remains enigmatic.